Anisotropic
5.3 Results and Discussion
5.2.1 Simulation Parameters
The calculations are carried out withE/σ0 = 100,m = 0.02(on the order of experimen- tally measured values of rate sensitivity in VACNTs [47]),˙0 = ˙ref,ν = 0.25, νp = 0.25.
The mesh geometry is that of a circular cylindrical pillar with an aspect ratio, H/R, of3.
The imposed velocity,vz, in Eq. (5.7) is fixed atκ˙refHwithκ = 0.004and the ramp time trise = 5/˙ref. The initial hardening portion of Eq. (5.5) was fixed at1 = 0.005, h1 = 5 throughout this study. Results are presented for variations in2,h2,h3.
The finite element mesh in all calculations consists of a uniform 80× 240 mesh of quadrilateral “crossed triangle” elements each of which isH/240×H/240.
If the analyses were quasi-static, these dimensionless parameters would be sufficient to characterize the formulation. However, dynamic, rather than quasi-static, analyses are carried out because, even though the response is generally quasi-static, dynamic snapping can occur due to the up-down-up shape ofg(p)(see Fig. 5.2). Hence, a density needs to be specified and is taken to beρ= 10−14σ0( ˙ref/H)2in non-dimensional form.
For σ0 = 0.1 MPa, H = 75 µm and ˙ref = 25 s−1, we have E = 10 MPa and ρ = 1.11×10−4 MPa/(m s)2 (ρ = 111 kg/m3). Also, vz = 7.5 µm/s and trise = 0.2, which corresponds to the applied displacement achieving its constant value when the over- all strain,uz(r, H, t)/H, reaches0.01. All of these values are of a similar order to those in the experiments.
Axial gradients in E and σ0 are incorporated into the material through multiplication of these variables by a dimensionless functionQ(z), whereQ(z) ≡ 1corresponds to the case in which there is no gradient withz evaluated at the center of the element for which the rescaledEandσ0values are being calculated.
regarding buckle characteristics (e.g., wavelength and amplitude) given a relationship be- tween the CNT microstructure and the hardening function. The qualitative results from an example set of parameters are summarized by Fig. 5.3 in which an experimental nominal stress,σn =P/A0, versus true strain,t = ln (1 +n), response from a uniaxial compres- sion experiment [44] is shown in Fig. 5.3a for reference. We plot the analogous response from a simulation (Fig. 5.3(b)) in terms of the nominal stress, σn = P/[πR2], and true strain, t = −ln [(H+uz(r, H, t))/H], where P is the sum of the nodal forces in the z direction at the top of the pillar. In both the experiment and the simulation, the area of the
b) a)
c)
Figure 5.3: Summary of buckle formation phenomena captured by simulations utilizing the proposed constitutive relation. a) Experimental data from a pillar microcompression [44]. Inset shows a closeup of the strain region from 0 to 0.3. b) The overall nominal stress, force/original area, and true stress, force/current area, versus true strain response from a simulated pillar undergoing periodic sequential buckling. Arrows mark the strains at which the outer displacement profiles are plotted in c). c)Outer displacement profiles corresponding to strains directly following a softening event and illustrating simultaneous buckle emergence.
top of the pillar is nearly constant during the process of sequential buckling so that the dif- ference between nominal and true stress is negligible over the majority of the range during which buckling occurs.
In both stress-strain responses in Fig. 5.3 there is a noticeable transition from the linear elastic region to the (sloped) plateau region. An inset of the experimental data is given in Fig. 5.3(a) in order to facilitate comparison with Fig. 5.3(b). The model gives similar behavior as illustrated by a comparison of the ratio of the stress at a strain of 0.2 to the stress at the beginning of the buckling regime (t = 0.05) in Fig. 5.3(a) to the same ratio for strains of 0.2 and 0.015 in Fig. 5.3(b). These ratios are around 1.3 in both cases. Periodic, local softening events occur within the plateaus and each corresponds to the formation of a new buckle. For the simulations, this finding is illustrated by a collection of curves showing the evolution of the outer surface,ur(R, z)/Rversusz/H, (Fig. 5.3(c)) at several discrete values of the overall pillar strain,t, that immediately follow a softening event. These strain values are indicated by arrows in Fig. 5.3(b) where both nominal and true (force/current area) stress values are shown. The outer displacement profiles clearly identify sequential buckle formation beginning at the bottom and progressing to the top. As in experiments, nearly all of the deformation is accommodated through the formation and evolution of localized buckles with the topmost region of the pillar remaining undeformed throughout, as evidenced in the strain contour plots in Fig. 5.3(c). All the simulation results in Fig.
5.3 correspond to the parameters defined in the caption to Fig. 5.2 and include a linear gradient, Q(z), that gives values of σ0 and E at z = H that are of 40% of their values atz = 0. Subsequently, we discuss the effect of an applied gradient on the overall pillar hardening.
The responses shown in Fig. 5.3 are approximately quasi-static since the total kinetic energy of the system remains around 2% of the input work throughout the short rise time discussed in Eq. (5.7), during which the structural response is largely elastic, and then drops to less than 1% for the remainder of the calculation.
A parameter study on the effect of strain rate is not carried out here. However, we have carried out calculations in which the strain rate is increased from the imposed value in Fig. 5.3 by a factor of 2 and decreased from that value by a factor of 1/2 (with all other
parameters fixed). As expected with a strain rate exponent of m = 0.02, the effect of changes in strain rate on the stress magnitude is small. The main effect is that the average stress drop that occurs with each buckling event is somewhat greater when the strain rate is doubled and somewhat less for the case when the strain rate is halved. This trend is in agreement with experiments carried out over 3.5 orders of magnitude in strain rate [44]
where it was observed that the magnitude of the stress undulations during buckling was greater at larger strain rates.
= 0.03 = 0.09 = 0.18 = 0.25
t t t t
r r r r
z
z
zz
p
Figure 5.4: Series of strain contour plots and deformed meshes clearly showing sequential deformation and the relatively undeformed upper region of the pillar.
In exploring the model’s parameter space, we found that the range in which buckle for- mation occurs, and where energy absorption is most effective, is limited. Within that buck- ling domain, we explore the separate contributions of the flow strength function’s ‘well’
width, formed by the intersection of the softening and rehardening slopes, and magnitude of the softening slope,h2, to changes in buckle morphology. Some calculations are carried out for a homogeneous pillar. However, based on images taken along the VACNT pillar height, there is reason to believe that there is an axial density gradient. Therefore, we in- vestigate the effect of gradients in the onset of plastic flow,σ0, and elastic modulus,E, for a single set of parameters. In order to illustrate the range of and reason for the limited buck- ling domain, a typical series of responses corresponding to selected flow strength functions, g(p), are shown in Fig. 5.5. All have fixed hardening slopes,h1 = 5andh3 = 1.5, with 1 = 0.005, but vary in the location of their minima, marked by the symbols in Fig. 5.5(a).
For example, a hardening function corresponding to a minimum at 2 = 0.1and 55% of
a) b)
d)
f) c)
e)
Figure 5.5: Influence of the ‘well’ minimum position on the formation and morphology of buckles. (h1 = 5, h3 = 1.5,1 = 0.005)a)Minima locations tested. Domains are denoted by (closed circles) buckling, (pluses) base-only buckling, (open circles) instability domi- nated, (diamond) base flow, and (squares) bulk flow. (b)–(f) Representative displacement profiles atr =R for overall strain levels oft = 0.05,0.10, 0.15, and0.20for each of the domains.
h/σ0(h2 = 5.0) exemplifies the buckling domain (filled circles) through the series of outer displacement profiles given for overall pillar strains oft = 0.05,0.10,0.15, and0.20(Fig.
5.5(a)). Outside of this buckling regime there are several types of behavior that can be roughly categorized into four groups. First, the instability domain (open circles), occurs where minima are located at similar strains but at a greater depth than that for the buck- ling domain, i.e., they possess a large softening slopeh2. Here, periodicity is completely lost and the deformation is dominated by local instability arising from the large magnitude of h2. Diagonally upward, at greater strain from the buckling domain, lies the base-only buckle domain (pluses). Here, the local instability due to softening is somewhat preserved, as evidenced by the small waves localized at the pillar base, however, the depth of the min- imum has decreased so much that the behavior begins to approach that of a typical foam, i.e., a hardening function in which the softening region is replaced by a flat line, h2 = 0.
Continuing toward minima at larger strain but similar depth, there is a bulk flow domain (open squares) where the the magnitude of (|h2|) has considerably decreased to the point that the presence of a local minimum has no noticeable contribution. Here, local flow is large and, as a result, the pillar undergoes extensive flow in a manner that is nearly identical to that seen for typical foam-like simulations (h2 = 0). Finally, holding the magnitude of h2approximately constant while extending minima to greater strains we enter what we call the base flow domain (open diamonds). In this domain, a large strain occurs only at the base of the pillar. Periodic buckles do not form as the extensive deformation, due to the large strain position of the minima, damps out any surface fluctuations that would form. It is noteworthy that all of the simulations shown in Fig. 5.5 were generated with no gradient, Q(z)≡1.
As a result of these observations of the different morphological domains and their de- pendence on the character of the flow strength, g(p), it becomes clear that a balance be- tween the magnitude ofh2 and the size of the ‘well’ ing(p)must exist in order to obtain the experimentally observed buckle morphology. In particular, it is evident that the local flow due to the ‘well’ size (i.e., width) must be limited enough that it does not wash out the undulations that form. Another way to state this is that the material must be strain con- strained. We interpret this within the known morphology of VACNTs by noting that the
intertube interactions (entanglement, van der Waals, etc.) limit the local strain that can be experienced by the material. The local softening captured byh2 arises from the high aspect ratio of the CNT struts which individually undergo a large drop in stiffness by buckling in uniaxial compression. We propose then that local strain constraint, combined with the high aspect ratio of the CNT struts, gives rise to the complex buckling behavior seen in so many VACNT compression experiments. It should be noted that only a variation in the extent of these domains is seen for changes in the value of the hardening coefficienth3 in Eq. (5.5) as variations in h3 give rise to similar domains that have the same relative positions with respect to each other.
Within the buckling domain, variations in buckle wavelength and amplitude can be decoupled from one another through control of specific characteristics of the hardening function. These findings are summarized in Fig. 5.6. We define ∆w as the plastic strain range from the value ofpat which the functiong(p)in Eq. (5.5) first attains a maximum to the value ofpat whichg(p)attains that value again as as illustrated in Fig. 5.6(a).∆w characterizes the width of the ‘well’ in the flow strength function,g(p). We find that with the depth of the minimum in g(p) and the value ofh2 held constant while varying ∆w, the wavelength of the buckles remains the same and the amplitude increases (Fig. 5.6(b)).
We quantify the changes in amplitude through a sine wave fit of the buckling region and define the relative change in amplitude,∆a, as
∆a= a∆−a0 a∆+a0
2
. (5.8)
wherea0 anda∆are the amplitudes determined from fits of the results from the reference parameters and from ±25% changes in ∆w, respectively. An analogous expression was used to quantify the relative changes in wavelength,∆λ. A relative change in amplitude of
−16% was obtained for a 25% decrease in∆wand a relative change of 12% was obtained for a 25% increase. The respective variations in wavelength of the buckles, ∆λ, were
−2% and 3%. Analogously, holding the depth of the minimum ofg(p)and∆w constant while varyingh2, the amplitude of the buckles is much less affected while the wavelength decreases (Fig. 5.6(d)). A sine wave fit revealed relative changes in∆λof 7% and−4% for a 25% decrease and a 25% increase inh2with∆svarying by−2% and 0.5% respectively.
25% variation
25% variation
h
constant
constant h
0.01
a)
c)
b)
d)
0.01
2
w
2
w
Figure 5.6: Variation in buckle wavelength and amplitude as a function of changes in∆w and in the magnitude ofh2. a)Schematic indicating the variations in the hardening func- tion, g(p), considered for 25% changes in ∆w. (red/squares = decreased, blue/circles
= increased)b) A series of displacement profiles at r = R corresponding to the harden- ing functions in a). An increased value of ∆w leads to increased buckle amplitude. c) Schematic indicating the variations in the hardening function for 25% changes in the mag- nitude ofh2. (red/squares = decreased, blue/circles = increased)d)A series of displacement profiles atr = Rcorresponding to the hardening functions in c). An increased magnitude ofh2leads to decreased buckle wavelength.
If h2 is fixed and ∆w varied (or vice versa), while the depth of the minimum in g(p) is allowed to change, the wavelength and amplitude variations are much more strongly coupled, and the buckle morphology varies in a way that precludes extracting a simple trend.
These correlations between the changes in h2 and∆w and the resulting variations in buckle wavelength and amplitude can be qualitatively related to real VACNT materials
as follows. If we presume that the ability of a VACNT material to flow is constrained by the tube-to-tube interactions, we expect that a more dense (number of tubes per unit volume) material would produce smaller amplitude buckles due to its increased number of interactions and therefore decreased deformability. If the magnitude of the negative slope, h2, is correlated with CNT strength in compression (i.e., smaller diameter tubes soften with a greater slope than larger diameter tubes), we expect that a material made with smaller tubes would have shorter wavelength buckles.
It has been speculated that the initiation of the sequential buckling phenomena observed experimentally by several research groups was due to an axial density gradient in the mate- rial [24, 44]. This is motivated by the observations of a lower CNT number density base of similarly grown VACNTs as discussed in Chapter 3 and Ref. [35]. An axial gradient in the number of load bearing members is expected to give rise to a corresponding inhomogene- ity in stiffness and yield stress. Since the precise correlation between the stiffness and the tube number density remains to be determined, we refer to it hereafter as a property gra- dient. We explore the effects of such a property gradient within the simulation framework by multiplication ofE andσ0 by the functionQ(z), where Q(z) = 1corresponds to the case in which there is no gradient. We find that changes in Young’s modulus,E, have very little effect on the shape of the stress-strain curve or the overall buckle morphology, but are included for completeness. We present the results of simulations with no gradient (black), 20% and 200% linear increases (blue/circles, red/squares), and a 10% linear decrease (green/triangles) in Q(z) in Fig. 5.7. Here, in Fig. 5.7(b) we plot curves of true stress, σt = P/[π(R+ur(R, H, t))2]versus true strain, t = −ln [(H+uz(r, H, t))/H]. Sim- ulations with no property gradient give rise to sequential periodic buckling (Fig. 5.7(b)), implying that buckle formation is robust against local density variations. The fixed con- straint of the rigid substrate, as is the case for as-grown CNT bundles, we model by the fixed displacement boundary conditions at z = 0. This constraint induces non-uniform deformation, which promotes buckle initiation at the base of the pillar. The gradient inσ0
has a marked effect on the hardening slope of the plateau region in the overall pillar stress- strain response as illustrated by the curves corresponding to 20 and 200% increases (Fig.
5.7(a)). Here, it is clear that the property gradient directly correlates with the overall pillar
hardening as a10×increase in property gradient yields an approximately 5× increase in overall strain hardening. For property gradients that are sufficiently large,∼ 400%, peri- odic buckling is no longer obtained. In our calculations, even a small reverse gradient can cause the sequential buckling to occur in the reverse direction (Fig. 5.7(c)), suggesting that a plausible explanation for variation in top-first vs. bottom-first buckling [43, 60] is the difference in the spatial location of the least number of load-bearing members within the sample.
Although our model captures some of the key qualitative features of VACNT pillar buckling, there are also some discrepancies between the model predictions and the exper- imental observations. One marked difference concerns the number and size of buckles.
This could be due to a number of idealizations including, for example: isotropy and the simple characterization of compressibility, such as the assumption of a constant value of the compressibility parameter,B, in Eq. (5.4). In addition, contact between buckles is not modeled.